 5demicube

Demipenteract
(5demicube)
Petrie polygon projectionType Uniform 5polytope Family (D_{n}) 5demicube Families (E_{n}) k_{21} polytope
1_{k2} polytopeCoxeter symbol 1_{21} Schläfli symbol {3^{1,2,1}}
h{4,3,3,3}
s{2,2,2,2}CoxeterDynkin diagram
4faces 26 10 {3^{1,1,1}}
16 {3,3,3}Cells 120 40 {3^{1,0,1}}
80 {3,3}Faces 160 {3} Edges 80 Vertices 16 Vertex figure
rectified 5cellPetrie polygon Octagon Symmetry group D_{5}, [3^{4,1,1}] = [1^{+},4,3^{3}]
[2^{4}]^{+}Properties convex In five dimensional geometry, a demipenteract or 5demicube is a semiregular 5polytope, constructed from a 5hypercube (penteract) with alternated vertices deleted.
It was discovered by Thorold Gosset. Since it was the only semiregular 5polytope (made of more than one type of regular hypercell), he called it a 5ic semiregular.
Coxeter named this polytope as 1_{21} from its CoxeterDynkin diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches. It exists in the k_{21} polytope family as 1_{21} with the Gosset polytopes: 2_{21}, 3_{21}, and 4_{21}.
Contents
Cartesian coordinates
Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:
 (±1,±1,±1,±1,±1)
with an odd number of plus signs.
Projected images
Perspective projection.Images
orthographic projections Coxeter plane B_{5} Graph Dihedral symmetry [10/2] Coxeter plane D_{5} D_{4} Graph Dihedral symmetry [8] [6] Coxeter plane D_{3} A_{3} Graph Dihedral symmetry [4] [4] Related polytopes
It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform polytera (uniform 5polytope) that can be constructed from the D_{5} symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.
t_{0}(1_{21})
t_{0,1}(1_{21})
t_{0,2}(1_{21})
t_{0,3}(1_{21})
t_{0,1,2}(1_{21})
t_{0,1,3}(1_{21})
t_{0,2,3}(1_{21})
t_{0,1,2,3}(1_{21})References
 T. Gosset: On the Regular and SemiRegular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5)
 H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 Richard Klitzing, 5D uniform polytopes (polytera), x3o3o *b3o3o  hin
External links
 Olshevsky, George, Demipenteract at Glossary for Hyperspace.
 Multidimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories: 5polytopes
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